[1]:

import numpy as np
import pandas as pd
from scipy.linalg import expm, null_space, solve
from scipy.sparse import csr_matrix
from scipy.sparse.linalg import expm_multiply, eigsh
import matplotlib.pyplot as plt
import networkx as nx
from networkx.algorithms.isomorphism import GraphMatcher
import pyvis.network as nt
import igraph

from qlinks.lattice.square_lattice import SquareLattice
from qlinks.symmetry.gauss_law import GaussLaw
from qlinks.solver.deep_first_search import DeepFirstSearch
from qlinks.computation_basis import ComputationBasis
from qlinks.model import QuantumLinkModel
from qlinks.visualizer.graph import GraphVisualizer

pd.set_option('display.max_rows', None)

[2]:

def format_custom_index(index):
    return [f"({i}) {idx}" for i, idx in enumerate(index)]

[3]:

def setup_dimer_model(lattice_shape, n_solution, coup_j, coup_rk):
    gauss_law = GaussLaw.from_staggered_charge_distri(*lattice_shape)
    gauss_law.flux_sector = (0, 0)
    dfs = DeepFirstSearch(gauss_law, max_steps=int(1e+8))
    basis = gauss_law.to_basis(dfs.solve(n_solution))
    model = QuantumLinkModel(coup_j, coup_rk, lattice_shape, basis)
    return basis, model

[4]:

def setup_link_model(lattice_shape, n_solution, coup_j, coup_rk):
    gauss_law = GaussLaw.from_zero_charge_distri(*lattice_shape)
    gauss_law.flux_sector = (0, 0)
    dfs = DeepFirstSearch(gauss_law, max_steps=int(1e+8))
    basis = gauss_law.to_basis(dfs.solve(n_solution))
    model = QuantumLinkModel(coup_j, coup_rk, lattice_shape, basis)
    return basis, model

QDM Lattice 4x2#

[9]:

coup_j, coup_rk = (1, 1)
basis, model = setup_dimer_model(lattice_shape=(4, 2), n_solution=16, coup_j=coup_j, coup_rk=coup_rk)
evals, evecs = np.linalg.eigh(model.hamiltonian)

2023-12-23 23:45:25 [deep_first_search.py] INFO: Deep First Search starts.
2023-12-23 23:45:25 [deep_first_search.py] INFO: Found 16 Solutions as required in 103 steps.

[10]:

df = basis.dataframe.copy(deep=True)
df.index = format_custom_index(df.index)
df

[10]:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
(0) 17595 0 1 0 0 0 1 0 0 1 0 1 1 1 0 1 1
(1) 18867 0 1 0 0 1 0 0 1 1 0 1 1 0 0 1 1
(2) 19638 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 0
(3) 36135 1 0 0 0 1 1 0 1 0 0 1 0 0 1 1 1
(4) 37947 1 0 0 1 0 1 0 0 0 0 1 1 1 0 1 1
(5) 39219 1 0 0 1 1 0 0 1 0 0 1 1 0 0 1 1
(6) 39990 1 0 0 1 1 1 0 0 0 0 1 1 0 1 1 0
(7) 44307 1 0 1 0 1 1 0 1 0 0 0 1 0 0 1 1
(8) 50283 1 1 0 0 0 1 0 0 0 1 1 0 1 0 1 1
(9) 51555 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1
(10) 52326 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 0
(11) 52773 1 1 0 0 1 1 1 0 0 0 1 0 0 1 0 1
(12) 55410 1 1 0 1 1 0 0 0 0 1 1 1 0 0 1 0
(13) 55857 1 1 0 1 1 0 1 0 0 0 1 1 0 0 0 1
(14) 60498 1 1 1 0 1 1 0 0 0 1 0 1 0 0 1 0
(15) 60945 1 1 1 0 1 1 1 0 0 0 0 1 0 0 0 1 
[11]:

evecs_df = pd.DataFrame.from_dict(
    {
        "eval": evals,
        "kin": [(evec.T @ model.kinetic_term @ evec).item() for evec in evecs.T],
        "pot": [(evec.T @ model.potential_term @ evec).item() for evec in evecs.T],
    }
)
evecs_df[evecs_df["kin"].abs() < 1e-12]

[11]:
eval kin pot
7 3.0 3.451266e-31 3.0
8 3.0 0.000000e+00 3.0
9 3.0 1.577722e-30 3.0 
[12]:

k_rank = np.linalg.matrix_rank(model.kinetic_term)
h_rank = np.linalg.matrix_rank(model.hamiltonian)
print(k_rank, h_rank, basis.n_states, (basis.n_states - k_rank) / basis.n_states)

10 15 16 0.375

[13]:

g = nx.from_numpy_array(-model.kinetic_term)
# assert nx.is_bipartite(g)
# left, right = nx.bipartite.sets(g)
# pos = nx.bipartite_layout(g, left, align="horizontal")
# pos = nx.spectral_layout(g)
# pos = {k: v + np.random.normal(0, 0.05, 2) for k, v in pos.items()}
# nx.draw(g, pos, with_labels=True, arrows=True, connectionstyle="arc3,rad=0.2")
# plt.show()

plt.spy(evecs, precision=1e-12, markersize=12)
plt.show()
../_images/notebooks_scar_robustness_9_0.png 
[124]:

G_pattern = nx.Graph()
G_pattern.add_nodes_from([0, 1, 2, 3, 4, 5, 6, 7, 8])
G_pattern.add_edges_from([(0, 1), (0, 2), (1, 3), (1, 4), (2, 4), (2, 5), (3, 6), (4, 6), (4, 7), (5, 7), (6, 8), (7, 8)])

# nx.draw(G_pattern, nx.spectral_layout(G_pattern), with_labels=True)
gm = GraphMatcher(g, G_pattern)
subgraphs = set([frozenset(subgraph_nodes) for subgraph_nodes in gm.subgraph_isomorphisms_iter()])
print(subgraphs)

{frozenset({3, 5, 7, 10, 11, 12, 13, 14, 15}), frozenset({0, 1, 2, 4, 5, 6, 8, 9, 10})}

[10]:

net = nt.Network(notebook=True, filter_menu=True, select_menu=True)
net.from_nx(g)
net.show_buttons(filter_=['nodes', 'edges', 'physics'])
net.show("nx.html")

Warning: When  cdn_resources is 'local' jupyter notebook has issues displaying graphics on chrome/safari. Use cdn_resources='in_line' or cdn_resources='remote' if you have issues viewing graphics in a notebook.
nx.html

[10]:

[13]:

ig = igraph.Graph.from_networkx(g)
ig.vs["label"] = [str(i) for i in range(ig.vcount())]
# color = ["orange" for i in range(ig.vcount())]
# highlight = [
#     [14, 13, 11, 7],
#     [1, 8, 2, 4],
# ]
# highlight_color = ["aqua", "yellowgreen"]
# for i, nodes in enumerate(highlight):
#     for node in nodes:
#         color[node] = highlight_color[i]
# ig.vs["color"] = color
# fig, ax = plt.subplots(figsize=(6, 6), facecolor="white")
igraph.plot(
    ig,
    layout=ig.layout_kamada_kawai(),
    vertex_size=16,
    vertex_color="orange",
    # target=ax
)

[13]:
../_images/notebooks_scar_robustness_12_0.svg
[14]:

entropies = np.asarray([model.entropy(evecs[:, i], 1, 0) for i in range(basis.n_states)])
entropy_df = pd.Series(entropies, name="entropy")
entropy_df

[14]:

0     1.536650
1     1.899841
2     0.695771
3     1.562885
4     1.528069
5     1.600664
6     1.932517
7     1.525501
8     1.611432
9     1.481633
10    1.941802
11    0.973698
12    1.147446
13    1.309708
14    1.502661
15    1.501048
Name: entropy, dtype: float64

[15]:

entropy_df.plot(style='o')

[15]:

<Axes: >
../_images/notebooks_scar_robustness_14_1.png 
[17]:

i = 8
plt.plot(evecs[:, i], linestyle='--', marker='o')
np.where(np.abs(evecs[:, i]) > 0.1)[0]

[17]:

array([ 1,  2,  4,  7,  8, 11, 13, 14])
../_images/notebooks_scar_robustness_15_1.png 
[19]:

fig, axes = plt.subplots(4, 4, figsize=(16, 16), facecolor="white")
ax = axes.flatten()

for i in range(basis.n_states):
    g = GraphVisualizer(SquareLattice(*model.shape, basis.dataframe.iloc[i]))
    g.plot(show=False, ax=ax[i], node_size=800)
    ax[i].set_title(f"{df.index[i]}")
../_images/notebooks_scar_robustness_16_0.png 
[21]:

coup_j = np.ones((np.prod(model.shape), 1))
coup_j[0] = 1.5
energy_lump_model = QuantumLinkModel(coup_j, coup_rk, model.shape, basis)
energy_lump_ham = energy_lump_model.hamiltonian

scar_state = np.zeros((basis.n_states, 1))
scar_state[np.array([1, 2, 4, 8]), 0] = np.array([1, 1, -1, -1])
scar_state /= np.linalg.norm(scar_state)

start, stop, num = 0, 1000, 100
evol_states = expm_multiply(-1j * energy_lump_ham, scar_state, start=start, stop=stop, num=num)
fidelity = [np.abs(evol_state.T @ scar_state).item() ** 2 for evol_state in evol_states]
entropy = [model.entropy(evol_state.flatten(), 1, 0) for evol_state in evol_states]

plt.plot(np.linspace(start, stop, num), fidelity, linestyle="--", marker="o")
plt.ylim(0.0001, 1.1)
# plt.yscale('log')
plt.xlabel("t")
plt.ylabel("Fidelity(t)")
plt.tight_layout()
plt.show()

plt.plot(np.linspace(start, stop, num), np.round(entropy, 12), linestyle="--", marker="o")
plt.xlabel("t")
plt.ylabel("EE(t)")
plt.tight_layout()
plt.show()

final_state = np.real_if_close(evol_states[-1], tol=1e-12)
print(f"final fidelity = {fidelity[-1]}")
print(f"O_kin @ psi_scar = {(model.kinetic_term @ final_state).flatten()}")
print(f"<O_kin> = {(final_state.T @ model.kinetic_term @ final_state).item()}")
print(f"<O_pot> = {(final_state.T @ model.potential_term @ final_state).item()}")
print(f"<H> = {(final_state.T @ model.hamiltonian @ final_state).item()}")
../_images/notebooks_scar_robustness_17_0.png
../_images/notebooks_scar_robustness_17_1.png 
final fidelity = 0.2163724158208147
O_kin @ psi_scar = [-0.244725  -0.2726498j   0.13566086-0.01666943j -0.05689253-0.01094762j
  0.16312646+0.07970323j -0.08939803+0.06206632j  0.1732982 +0.11400362j
  0.02198015-0.14927196j -0.00661834+0.06313454j  0.16816636-0.08968338j
 -0.26670515-0.12337784j -0.09177027-0.22724696j  0.02588716-0.0098794j
  0.16312646+0.07970323j -0.00661834+0.06313454j  0.02588716-0.0098794j
  0.32625293+0.15940645j]
<O_kin> = (0.12547370021383802-0.04254736654748517j)
<O_pot> = (0.7340468683168595+2.010551039444004j)
<H> = (0.8595205685306977+1.9680036728965187j)

QDM Lattice 4x4#

[22]:

coup_j, coup_rk = (1, 1)
basis, model = setup_dimer_model(lattice_shape=(4, 4), n_solution=132, coup_j=coup_j, coup_rk=coup_rk)
evals, evecs = np.linalg.eigh(model.hamiltonian)

2023-12-23 23:48:46 [deep_first_search.py] INFO: Deep First Search starts.
2023-12-23 23:48:46 [deep_first_search.py] INFO: Found 132 Solutions as required in 1626 steps.

[23]:

df = basis.dataframe.copy(deep=True)
df.index = format_custom_index(df.index)
df

[23]:
0 1 2 3 4 5 6 7 8 9 ... 22 23 24 25 26 27 28 29 30 31
(0) 1142025915 0 1 0 0 0 1 0 0 0 0 ... 1 0 1 0 1 1 1 0 1 1
(1) 1143328443 0 1 0 0 0 1 0 0 0 0 ... 1 0 1 0 1 1 1 0 1 1
(2) 1146285243 0 1 0 0 0 1 0 0 0 1 ... 0 0 1 0 1 1 1 0 1 1
(3) 1147587771 0 1 0 0 0 1 0 0 0 1 ... 0 0 1 0 1 1 1 0 1 1
(4) 1147913403 0 1 0 0 0 1 0 0 0 1 ... 0 0 1 0 1 1 1 0 1 1
(5) 1147914675 0 1 0 0 0 1 0 0 0 1 ... 0 1 1 0 1 1 0 0 1 1
(6) 1152797883 0 1 0 0 0 1 0 0 1 0 ... 0 0 1 0 1 1 1 0 1 1
(7) 1152818235 0 1 0 0 0 1 0 0 1 0 ... 0 0 0 0 1 1 1 0 1 1
(8) 1153123515 0 1 0 0 0 1 0 0 1 0 ... 0 0 1 0 1 1 1 0 1 1
(9) 1153124787 0 1 0 0 0 1 0 0 1 0 ... 0 1 1 0 1 1 0 0 1 1
(10) 1153143867 0 1 0 0 0 1 0 0 1 0 ... 0 0 0 0 1 1 1 0 1 1
(11) 1153145139 0 1 0 0 0 1 0 0 1 0 ... 0 1 0 0 1 1 0 0 1 1
(12) 1153150227 0 1 0 0 0 1 0 0 1 0 ... 0 1 0 0 0 1 0 0 1 1
(13) 1153161777 0 1 0 0 0 1 0 0 1 0 ... 1 0 0 0 1 1 0 0 0 1
(14) 1153166865 0 1 0 0 0 1 0 0 1 0 ... 1 0 0 0 0 1 0 0 0 1
(15) 1231275195 0 1 0 0 1 0 0 1 0 1 ... 0 0 1 0 1 1 1 0 1 1
(16) 1231276467 0 1 0 0 1 0 0 1 0 1 ... 0 1 1 0 1 1 0 0 1 1
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132 rows × 32 columns

[24]:

evecs_df = pd.DataFrame.from_dict(
    {
        "eval": evals,
        "kin": [(evec.T @ model.kinetic_term @ evec).item() for evec in evecs.T],
        "pot": [(evec.T @ model.potential_term @ evec).item() for evec in evecs.T],
    }
)
evecs_df[evecs_df["kin"].abs() < 1e-12]

[24]:
eval kin pot
47 4.0 -9.907176e-30 4.0
48 4.0 -8.923989e-30 4.0
49 4.0 -9.959369e-30 4.0
50 4.0 -6.914859e-30 4.0
51 4.0 -5.756219e-30 4.0
52 4.0 3.056836e-30 4.0
53 4.0 -2.465190e-30 4.0
54 4.0 -1.015658e-29 4.0
55 4.0 -1.005798e-29 4.0 
[25]:

k_rank = np.linalg.matrix_rank(model.kinetic_term)
h_rank = np.linalg.matrix_rank(model.hamiltonian)
print(k_rank, h_rank, basis.n_states, (basis.n_states - k_rank) / basis.n_states)

102 131 132 0.22727272727272727

[26]:

g = nx.from_numpy_array(-model.kinetic_term)
# left, right = nx.bipartite.sets(g)
# pos = nx.bipartite_layout(g, left, align='horizontal')
# pos = nx.spectral_layout(g)
# pos = {k: v + np.random.normal(0, 0.01, 2) for k, v in pos.items()}
# fig = plt.figure(figsize=(82, 32))
# nx.draw(g, pos, with_labels=True, node_size=5000, font_size=42, arrows=True, connectionstyle="arc3,rad=0.15")
# plt.show()

plt.spy(evecs, precision=1e-12, markersize=0.1)
plt.show()
../_images/notebooks_scar_robustness_23_0.png 
[167]:

# G_pattern = nx.Graph()
# G_pattern.add_nodes_from([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14])
# G_pattern.add_edges_from([(0, 1), (0, 5), (2, 1), (2, 7), (1, 3), (1, 4), (3, 5), (3, 6), (4, 6), (4, 7), (5, 10), (5, 9), (6, 8), (6, 9), (7, 8), (7, 12), (8, 11), (9, 11), (10, 11), (12, 11), (3, 13), (13, 9), (4, 14), (14, 8)])
G_pattern = g.subgraph([131, 94, 116, 130, 129, 93, 92, 115, 114, 85, 84, 91, 128, 83, 82, 113])
# print(np.linalg.matrix_rank(nx.to_numpy_array(G_pattern)))

# left, right = nx.bipartite.sets(G_pattern)
nx.draw(G_pattern, nx.spectral_layout(G_pattern), with_labels=True)
gm = GraphMatcher(g, G_pattern)
subgraphs = set([frozenset(subgraph_nodes) for subgraph_nodes in gm.subgraph_isomorphisms_iter()])
print(len(subgraphs))
for subgraph in subgraphs:
    if any(head_node in list(subgraph) for head_node in [131, 0, 54, 76, 97, 35, 14, 117]):
        pass
    else:
        print(subgraph)

12
frozenset({3, 4, 69, 70, 6, 8, 72, 73, 82, 83, 84, 85, 27, 28, 29, 31})
frozenset({102, 103, 104, 105, 49, 50, 51, 52, 120, 121, 122, 123, 60, 61, 62, 63})
frozenset({128, 79, 80, 81, 82, 87, 89, 90, 91, 109, 110, 112, 113, 123, 125, 127})
frozenset({8, 9, 10, 11, 17, 18, 19, 20, 39, 40, 41, 42, 46, 47, 48, 49})
../_images/notebooks_scar_robustness_24_1.png 
[52]:

net = nt.Network(notebook=True, filter_menu=True, select_menu=True, font_color="black")
net.from_nx(g)
for node in net.nodes:
    node["title"] = str(node["id"])
net.show_buttons(filter_=['nodes', 'edges', 'physics'])
net.show("nx.html")

Warning: When  cdn_resources is 'local' jupyter notebook has issues displaying graphics on chrome/safari. Use cdn_resources='in_line' or cdn_resources='remote' if you have issues viewing graphics in a notebook.
nx.html

[52]:

[31]:

ig = igraph.Graph.from_networkx(g)
ig.vs["label"] = [str(i) for i in range(ig.vcount())]
# color = ["orange" for i in range(ig.vcount())]
# highlight = [
#     [34, 65, 86, 124],
#     [96, 108, 107, 126],
#     [93, 115, 92, 114],
#     [77, 74, 15, 5],
#     [59, 101, 58, 100],
#     [25, 67, 26, 68],
#     [55, 37, 32, 7],
#     [43, 21, 44, 22]
# ]
# highlight_color = ["aqua", "deepskyblue", "yellowgreen", "royalblue", "blueviolet", "cornflowerblue", "limegreen", "pink"]
# for i, nodes in enumerate(highlight):
#     for node in nodes:
#         color[node] = highlight_color[i]
# ig.vs["color"] = color
# fig, ax = plt.subplots(figsize=(10, 10), facecolor="white")
igraph.plot(
    ig,
    layout=ig.layout_kamada_kawai(),
    vertex_size=12,
    vertex_color="orange",
    vertex_label_size=8,
    # target=ax
)

[31]:
../_images/notebooks_scar_robustness_26_0.svg
[32]:

entropies = np.asarray([model.entropy(evecs[:, i], 2, 1) for i in range(basis.n_states)])
entropy_df = pd.Series(entropies, name="entropy")
entropy_df[entropy_df < 1.9]

[32]:

38     1.863804
41     1.898850
51     1.872677
53     1.834227
54     1.804352
55     1.807246
112    1.827273
123    1.881107
Name: entropy, dtype: float64

[33]:

entropy_df.plot(style='o')

[33]:

<Axes: >
../_images/notebooks_scar_robustness_28_1.png 
[35]:

i = 50
plt.plot(evecs[:, i], linestyle='--', marker='o')
np.where(np.abs(evecs[:, i]) > 0.2)[0]

[35]:

array([ 25,  26,  67,  68,  96, 107, 108, 126])
../_images/notebooks_scar_robustness_29_1.png 
[36]:

# {i: len(np.where(np.abs(evecs[:, i]) > 1e-12)[0]) for i in range(300)}

fig, axes = plt.subplots(2, 2, figsize=(16, 12), facecolor="white")
ax = axes.flatten()

for i, val in enumerate([25, 26, 67, 68]):
    g = GraphVisualizer(SquareLattice(*model.shape, basis.dataframe.iloc[val]))
    g.plot(show=False, ax=ax[i], node_size=800)
    ax[i].set_title(f"{df.index[val]}")
../_images/notebooks_scar_robustness_30_0.png 
[40]:

coup_j = np.ones((np.prod(model.shape), 1))
coup_j[5] = 1.1
energy_lump_model = QuantumLinkModel(coup_j, coup_rk, model.shape, basis)
energy_lump_ham = energy_lump_model.hamiltonian

scar_state = np.zeros((basis.n_states, 1))
scar_state[np.array([25, 26, 67, 68]), 0] = np.array([1, -1, -1, 1])
scar_state /= np.linalg.norm(scar_state)

start, stop, num = 0, 200, 100
evol_states = expm_multiply(-1j * energy_lump_ham, scar_state, start=start, stop=stop, num=num)
fidelity = [np.abs(evol_state.T @ scar_state).item() ** 2 for evol_state in evol_states]
entropy = [model.entropy(evol_state.flatten(), 2, 0) for evol_state in evol_states]

plt.plot(np.linspace(start, stop, num), fidelity, linestyle="--", marker="o")
plt.ylim(0.0001, 1.1)
# plt.yscale('log')
plt.xlabel("t")
plt.ylabel("Fidelity(t)")
plt.tight_layout()
plt.show()

plt.plot(np.linspace(start, stop, num), np.round(entropy, 12), linestyle="--", marker="o")
plt.xlabel("t")
plt.ylabel("EE(t)")
plt.tight_layout()
plt.show()

final_state = np.real_if_close(evol_states[-1], tol=1e-12)
print(f"final fidelity = {fidelity[-1]}")
print(f"O_kin @ psi_scar = {(model.kinetic_term @ final_state).flatten()}")
print(f"<O_kin> = {(final_state.T @ model.kinetic_term @ final_state).item()}")
print(f"<O_pot> = {(final_state.T @ model.potential_term @ final_state).item()}")
print(f"<H> = {(final_state.T @ model.hamiltonian @ final_state).item()}")
../_images/notebooks_scar_robustness_31_0.png
../_images/notebooks_scar_robustness_31_1.png 
final fidelity = 0.9879392050588764
O_kin @ psi_scar = [-1.90812800e-16+8.28091669e-17j -3.65133183e-16+1.00329721e-15j
 -2.35776161e-16+4.67119381e-16j -7.80625564e-18+4.16333634e-17j
 -1.74795127e-02-9.03285214e-03j -2.73938464e-03+5.20666229e-03j
  1.74795127e-02+9.03285214e-03j  2.73938464e-03-5.20666229e-03j
 -3.22658567e-16+4.72712147e-17j -3.61119152e-03+6.08438797e-03j
  3.61119152e-03-6.08438797e-03j -2.89698820e-16-2.24646690e-16j
  3.85072889e-17-4.05411804e-16j -4.56055529e-17-8.64761428e-17j
 -2.22121645e-16-2.27589252e-16j -2.73938464e-03+5.20666229e-03j
  1.20663790e-02-7.38583877e-03j -3.61119152e-03+6.08438797e-03j
 -4.42050954e-04-4.97435852e-03j -8.76035355e-17-4.77048956e-17j
 -2.18963565e-03-5.76980809e-03j  5.35393878e-03-1.66115198e-03j
  5.35393878e-03-1.66115198e-03j -2.48235535e-03-7.62830643e-03j
  3.91205599e-02+1.00014163e-03j -2.45225784e-03-5.36830970e-03j
 -2.45225784e-03-5.36830970e-03j -1.81588903e-02+1.32788326e-02j
 -1.15137815e-02-1.60958899e-02j -1.52035833e-03+1.38019956e-02j
 -6.87649577e-03+3.88407163e-03j -3.55428138e-02-4.03318656e-03j
 -8.13810349e-03+4.73364612e-03j  6.22652007e-16-3.35461204e-18j
  1.36900345e-17+2.68423982e-16j  1.23423291e-16-3.71341662e-17j
  1.26827388e-16+1.89787073e-16j  2.73938464e-03-5.20666229e-03j
 -1.20663790e-02+7.38583877e-03j  3.61119152e-03-6.08438797e-03j
 -5.81132364e-17-1.56125113e-16j  4.42050954e-04+4.97435852e-03j
  2.18963565e-03+5.76980809e-03j -5.35393878e-03+1.66115198e-03j
 -5.35393878e-03+1.66115198e-03j  2.48235535e-03+7.62830643e-03j
 -2.87096735e-16-2.51534904e-16j -2.18963565e-03-5.76980809e-03j
  2.18963565e-03+5.76980809e-03j  1.34007389e-16-2.28116137e-16j
  2.75547687e-16-3.17039042e-16j  4.45518215e-16+1.96542943e-16j
  1.43290475e-16-2.00399236e-16j -6.87649577e-03+3.88407163e-03j
 -5.44204673e-04+1.14773075e-03j -8.13810349e-03+4.73364612e-03j
  2.35990715e-02-1.67861370e-02j  7.02563008e-17-4.23706209e-16j
  5.35393878e-03-1.66115198e-03j -5.35393878e-03+1.66115198e-03j
  3.03576608e-16-3.64725611e-16j  5.39771104e-16+2.56207448e-17j
  1.80902577e-16-6.46329022e-17j -8.02520617e-17-6.93554591e-17j
 -3.40951184e-16-5.82888883e-16j -7.08834627e-17-3.55703350e-16j
 -3.91205599e-02-1.00014163e-03j  2.45225784e-03+5.36830970e-03j
  2.45225784e-03+5.36830970e-03j  1.81588903e-02-1.32788326e-02j
  1.52035833e-03-1.38019956e-02j  6.87649577e-03-3.88407163e-03j
  1.15137815e-02+1.60958899e-02j  3.55428138e-02+4.03318656e-03j
  8.13810349e-03-4.73364612e-03j  6.87649577e-03-3.88407163e-03j
  5.44204673e-04-1.14773075e-03j  8.13810349e-03-4.73364612e-03j
 -2.35990715e-02+1.67861370e-02j  3.67784958e-16+6.57586511e-16j
 -3.25471719e-17+8.81681133e-16j  6.16651591e-17+1.59015677e-16j
  3.14635470e-16+2.12937307e-16j -1.18140254e-02+1.28552941e-03j
  1.18140254e-02-1.28552941e-03j -1.80844922e-16+8.41340886e-17j
  1.78615569e-17+2.70011768e-16j -4.65274673e-17+9.34849019e-16j
  1.29367576e-16+1.97524062e-16j  2.64508169e-16+3.24349044e-16j
  4.82911320e-16+5.76082182e-16j  6.27969898e-16-4.72712147e-17j
  6.81219495e-03+3.32206124e-03j -6.81219495e-03-3.32206124e-03j
  7.38992201e-16-3.16803875e-16j -1.39311762e-16-3.23453896e-16j
  5.11433383e-16+1.12182132e-16j -4.72098495e-17+1.51703847e-16j
  2.51195966e-16-1.93511765e-16j -1.12757026e-17-1.14925430e-16j
  5.35393878e-03-1.66115198e-03j -5.35393878e-03+1.66115198e-03j
  4.81819445e-16+1.53956708e-16j  1.95598105e-16-3.36710327e-17j
  9.04506532e-17+1.25506524e-17j  1.63631090e-16+1.55247123e-16j
  6.91157282e-16+1.26216929e-16j -3.56291129e-16+5.00484060e-16j
  4.91557507e-16+6.71710244e-17j -4.64753718e-16-1.30394279e-16j
 -1.16895634e-16-4.46616720e-18j  2.44015932e-16-1.82750591e-16j
 -7.60001168e-17+2.25361320e-16j  5.22151766e-16+3.44559450e-16j
  6.81219495e-03+3.32206124e-03j -6.81219495e-03-3.32206124e-03j
  1.33573708e-16+6.41630846e-16j -2.94469310e-16-2.01227923e-16j
 -2.48235535e-03-7.62830643e-03j  2.48235535e-03+7.62830643e-03j
  1.46584134e-16-5.63785130e-17j -4.65731334e-17-1.08286819e-16j
  1.82031844e-16+1.03653535e-16j  6.03103098e-18+3.67393821e-16j
 -6.67119402e-17-3.54115563e-16j -2.58698978e-16+2.47121582e-17j
 -3.76167004e-16+4.55472952e-16j  6.00887640e-16+3.73918371e-16j
 -2.15105711e-16+1.77809156e-16j  1.95943013e-02+2.45828327e-03j
 -1.95943013e-02-2.45828327e-03j  2.50667542e-16+1.77158635e-16j]
<O_kin> = (-0.001676203291806102-0.0023445400116498074j)
<O_pot> = (3.917733396220584-0.663732907448769j)
<H> = (3.916057192928778-0.666077447460419j)

QDM Lattice 6x2#

[20]:

coup_j, coup_rk = (1, 1)
basis, model = setup_dimer_model(lattice_shape=(6, 2), n_solution=76, coup_j=coup_j, coup_rk=coup_rk)
evals, evecs = np.linalg.eigh(model.hamiltonian)

2023-12-23 21:48:24 [deep_first_search.py] INFO: Deep First Search starts.
2023-12-23 21:48:24 [deep_first_search.py] INFO: Found 76 Solutions as required in 693 steps.

[21]:

df = basis.dataframe.copy(deep=True)
df.index = format_custom_index(df.index)
df

[21]:
0 1 2 3 4 5 6 7 8 9 ... 14 15 16 17 18 19 20 21 22 23
(0) 4475835 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(1) 4496307 0 1 0 0 0 1 0 0 1 0 ... 1 1 1 0 1 1 0 0 1 1
(2) 4508598 0 1 0 0 0 1 0 0 1 1 ... 1 1 1 0 1 1 0 1 1 0
(3) 4774695 0 1 0 0 1 0 0 0 1 1 ... 1 1 0 0 1 0 0 1 1 1
(4) 4803387 0 1 0 0 1 0 0 1 0 1 ... 1 1 0 0 1 1 1 0 1 1
(5) 4823859 0 1 0 0 1 0 0 1 1 0 ... 1 1 0 0 1 1 0 0 1 1
(6) 4836150 0 1 0 0 1 0 0 1 1 1 ... 1 1 0 0 1 1 0 1 1 0
(7) 4905747 0 1 0 0 1 0 1 0 1 1 ... 1 1 0 0 0 1 0 0 1 1
(8) 5000043 0 1 0 0 1 1 0 0 0 1 ... 1 1 0 1 1 0 1 0 1 1
(9) 5020515 0 1 0 0 1 1 0 0 1 0 ... 1 1 0 1 1 0 0 0 1 1
(10) 5032806 0 1 0 0 1 1 0 0 1 1 ... 1 1 0 1 1 0 0 1 1 0
(11) 5081970 0 1 0 0 1 1 0 1 1 0 ... 1 1 0 1 1 1 0 0 1 0
(12) 5163858 0 1 0 0 1 1 1 0 1 1 ... 1 1 0 1 0 1 0 0 1 0
(13) 8704695 1 0 0 0 0 1 0 0 1 1 ... 1 0 1 0 1 1 0 1 1 1
(14) 9032247 1 0 0 0 1 0 0 1 1 1 ... 1 0 0 0 1 1 0 1 1 1
(15) 9228903 1 0 0 0 1 1 0 0 1 1 ... 1 0 0 1 1 0 0 1 1 1
(16) 9257595 1 0 0 0 1 1 0 1 0 1 ... 1 0 0 1 1 1 1 0 1 1
(17) 9278067 1 0 0 0 1 1 0 1 1 0 ... 1 0 0 1 1 1 0 0 1 1
(18) 9290358 1 0 0 0 1 1 0 1 1 1 ... 1 0 0 1 1 1 0 1 1 0
(19) 9359955 1 0 0 0 1 1 1 0 1 1 ... 1 0 0 1 0 1 0 0 1 1
(20) 9716667 1 0 0 1 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(21) 9737139 1 0 0 1 0 1 0 0 1 0 ... 1 1 1 0 1 1 0 0 1 1
(22) 9749430 1 0 0 1 0 1 0 0 1 1 ... 1 1 1 0 1 1 0 1 1 0
(23) 10015527 1 0 0 1 1 0 0 0 1 1 ... 1 1 0 0 1 0 0 1 1 1
(24) 10044219 1 0 0 1 1 0 0 1 0 1 ... 1 1 0 0 1 1 1 0 1 1
(25) 10064691 1 0 0 1 1 0 0 1 1 0 ... 1 1 0 0 1 1 0 0 1 1
(26) 10076982 1 0 0 1 1 0 0 1 1 1 ... 1 1 0 0 1 1 0 1 1 0
(27) 10146579 1 0 0 1 1 0 1 0 1 1 ... 1 1 0 0 0 1 0 0 1 1
(28) 10240875 1 0 0 1 1 1 0 0 0 1 ... 1 1 0 1 1 0 1 0 1 1
(29) 10261347 1 0 0 1 1 1 0 0 1 0 ... 1 1 0 1 1 0 0 0 1 1
(30) 10273638 1 0 0 1 1 1 0 0 1 1 ... 1 1 0 1 1 0 0 1 1 0
(31) 10322802 1 0 0 1 1 1 0 1 1 0 ... 1 1 0 1 1 1 0 0 1 0
(32) 10404690 1 0 0 1 1 1 1 0 1 1 ... 1 1 0 1 0 1 0 0 1 0
(33) 11325735 1 0 1 0 1 1 0 0 1 1 ... 0 1 0 0 1 0 0 1 1 1
(34) 11354427 1 0 1 0 1 1 0 1 0 1 ... 0 1 0 0 1 1 1 0 1 1
(35) 11374899 1 0 1 0 1 1 0 1 1 0 ... 0 1 0 0 1 1 0 0 1 1
(36) 11387190 1 0 1 0 1 1 0 1 1 1 ... 0 1 0 0 1 1 0 1 1 0
(37) 11456787 1 0 1 0 1 1 1 0 1 1 ... 0 1 0 0 0 1 0 0 1 1
(38) 12863163 1 1 0 0 0 1 0 0 0 1 ... 1 0 1 0 1 1 1 0 1 1
(39) 12883635 1 1 0 0 0 1 0 0 1 0 ... 1 0 1 0 1 1 0 0 1 1
(40) 12895926 1 1 0 0 0 1 0 0 1 1 ... 1 0 1 0 1 1 0 1 1 0
(41) 12903093 1 1 0 0 0 1 0 0 1 1 ... 1 0 1 0 1 1 0 1 0 1
(42) 13162023 1 1 0 0 1 0 0 0 1 1 ... 1 0 0 0 1 0 0 1 1 1
(43) 13190715 1 1 0 0 1 0 0 1 0 1 ... 1 0 0 0 1 1 1 0 1 1
(44) 13211187 1 1 0 0 1 0 0 1 1 0 ... 1 0 0 0 1 1 0 0 1 1
(45) 13223478 1 1 0 0 1 0 0 1 1 1 ... 1 0 0 0 1 1 0 1 1 0
(46) 13230645 1 1 0 0 1 0 0 1 1 1 ... 1 0 0 0 1 1 0 1 0 1
(47) 13293075 1 1 0 0 1 0 1 0 1 1 ... 1 0 0 0 0 1 0 0 1 1
(48) 13387371 1 1 0 0 1 1 0 0 0 1 ... 1 0 0 1 1 0 1 0 1 1
(49) 13407843 1 1 0 0 1 1 0 0 1 0 ... 1 0 0 1 1 0 0 0 1 1
(50) 13420134 1 1 0 0 1 1 0 0 1 1 ... 1 0 0 1 1 0 0 1 1 0
(51) 13427301 1 1 0 0 1 1 0 0 1 1 ... 1 0 0 1 1 0 0 1 0 1
(52) 13469298 1 1 0 0 1 1 0 1 1 0 ... 1 0 0 1 1 1 0 0 1 0
(53) 13476465 1 1 0 0 1 1 0 1 1 0 ... 1 0 0 1 1 1 0 0 0 1
(54) 13551186 1 1 0 0 1 1 1 0 1 1 ... 1 0 0 1 0 1 0 0 1 0
(55) 13558353 1 1 0 0 1 1 1 0 1 1 ... 1 0 0 1 0 1 0 0 0 1
(56) 13928370 1 1 0 1 0 1 0 0 1 0 ... 1 1 1 0 1 1 0 0 1 0
(57) 13935537 1 1 0 1 0 1 0 0 1 0 ... 1 1 1 0 1 1 0 0 0 1
(58) 14173995 1 1 0 1 1 0 0 0 0 1 ... 1 1 0 0 1 0 1 0 1 1
(59) 14194467 1 1 0 1 1 0 0 0 1 0 ... 1 1 0 0 1 0 0 0 1 1
(60) 14206758 1 1 0 1 1 0 0 0 1 1 ... 1 1 0 0 1 0 0 1 1 0
(61) 14213925 1 1 0 1 1 0 0 0 1 1 ... 1 1 0 0 1 0 0 1 0 1
(62) 14255922 1 1 0 1 1 0 0 1 1 0 ... 1 1 0 0 1 1 0 0 1 0
(63) 14263089 1 1 0 1 1 0 0 1 1 0 ... 1 1 0 0 1 1 0 0 0 1
(64) 14337810 1 1 0 1 1 0 1 0 1 1 ... 1 1 0 0 0 1 0 0 1 0
(65) 14344977 1 1 0 1 1 0 1 0 1 1 ... 1 1 0 0 0 1 0 0 0 1
(66) 14452578 1 1 0 1 1 1 0 0 1 0 ... 1 1 0 1 1 0 0 0 1 0
(67) 14459745 1 1 0 1 1 1 0 0 1 0 ... 1 1 0 1 1 0 0 0 0 1
(68) 15484203 1 1 1 0 1 1 0 0 0 1 ... 0 1 0 0 1 0 1 0 1 1
(69) 15504675 1 1 1 0 1 1 0 0 1 0 ... 0 1 0 0 1 0 0 0 1 1
(70) 15516966 1 1 1 0 1 1 0 0 1 1 ... 0 1 0 0 1 0 0 1 1 0
(71) 15524133 1 1 1 0 1 1 0 0 1 1 ... 0 1 0 0 1 0 0 1 0 1
(72) 15566130 1 1 1 0 1 1 0 1 1 0 ... 0 1 0 0 1 1 0 0 1 0
(73) 15573297 1 1 1 0 1 1 0 1 1 0 ... 0 1 0 0 1 1 0 0 0 1
(74) 15648018 1 1 1 0 1 1 1 0 1 1 ... 0 1 0 0 0 1 0 0 1 0
(75) 15655185 1 1 1 0 1 1 1 0 1 1 ... 0 1 0 0 0 1 0 0 0 1

76 rows × 24 columns

[23]:

evecs_df = pd.DataFrame.from_dict(
    {
        "eval": evals,
        "kin": [(evec.T @ model.kinetic_term @ evec).item() for evec in evecs.T],
        "pot": [(evec.T @ model.potential_term @ evec).item() for evec in evecs.T],
    }
)
evecs_df[evecs_df["kin"].abs() < 1e-12]

[23]:
eval kin pot
34 4.0 -2.081668e-16 4.0
36 4.0 -3.712308e-16 4.0
37 4.0 1.013079e-15 4.0
38 4.0 4.718448e-16 4.0
39 4.0 -6.383782e-16 4.0
40 4.0 7.979728e-16 4.0
41 4.0 1.483189e-16 4.0 
[22]:

k_rank = np.linalg.matrix_rank(model.kinetic_term)
h_rank = np.linalg.matrix_rank(model.hamiltonian)
print(k_rank, h_rank, basis.n_states, (basis.n_states - k_rank) / basis.n_states)

60 75 76 0.21052631578947367

[6]:

fig, axes = plt.subplots(19, 4, figsize=(30, 60), facecolor="white")
ax = axes.flatten()

for i in range(basis.n_states):
    g = GraphVisualizer(SquareLattice(*model.shape, basis.dataframe.iloc[i]))
    g.plot(show=False, ax=ax[i], node_size=800)
    ax[i].set_title(f"{df.index[i]}")
../_images/notebooks_scar_robustness_37_0.png 
[24]:

g = nx.from_numpy_array(-model.kinetic_term)
# left, right = nx.bipartite.sets(g)
# pos = nx.bipartite_layout(g, left, align='horizontal')
# pos = nx.spectral_layout(g)
# pos = {k: v + np.random.normal(0, 0.03, 2) for k, v in pos.items()}
# fig = plt.figure(figsize=(82, 32))
# nx.draw(g, pos, with_labels=True, node_size=5000, font_size=42, arrows=True, connectionstyle="arc3,rad=0.15")
# plt.show()

plt.spy(evecs, precision=1e-12, markersize=1)
plt.show()
../_images/notebooks_scar_robustness_38_0.png 
[95]:

net = nt.Network(notebook=True, filter_menu=True, select_menu=True, font_color="black")
net.from_nx(g)
for node in net.nodes:
    node["title"] = str(node["id"])
net.show_buttons(filter_=['nodes', 'edges', 'physics'])
net.show("nx.html")

Warning: When  cdn_resources is 'local' jupyter notebook has issues displaying graphics on chrome/safari. Use cdn_resources='in_line' or cdn_resources='remote' if you have issues viewing graphics in a notebook.
nx.html

[95]:

[81]:

ig = igraph.Graph.from_networkx(g)
fig, ax = plt.subplots(figsize=(10, 8), facecolor="white")
igraph.plot(ig, layout=ig.layout_kamada_kawai(), vertex_size=10, vertex_color="orange", target=ax)

[81]:

<igraph.drawing.matplotlib.graph.GraphArtist at 0x13210e350>
../_images/notebooks_scar_robustness_40_1.png 
[121]:

entropies = np.asarray([model.entropy(evecs[:, i], 0, 0) for i in range(basis.n_states)])
entropy_df = pd.Series(entropies, name="entropy")
entropy_df.loc[(entropy_df.index >= 20) & (entropy_df.index <= 50) & (entropy_df < 1.75)]

[121]:

20    1.058436
21    1.059513
22    1.061746
23    1.054004
24    1.074146
25    1.075756
26    1.012101
27    1.045869
28    1.041082
29    1.036557
30    0.997521
31    1.003358
32    1.054331
33    1.045045
34    1.032524
35    1.043826
36    1.053413
37    1.049003
38    1.061464
39    1.071072
40    1.050495
41    1.034524
42    1.093099
43    1.087584
44    0.949660
45    1.050145
46    1.028086
47    1.045876
48    1.041796
49    1.025918
50    1.020980
Name: entropy, dtype: float64

[122]:

entropy_df.plot(style='o')

[122]:

<Axes: >
../_images/notebooks_scar_robustness_42_1.png 
[27]:

i = 41
plt.plot(evecs[:, i], linestyle='--', marker='o')
np.where(np.abs(evecs[:, i]) > 0.15)[0]

[27]:

array([ 7, 12, 19, 23, 29, 30, 33, 44, 49, 52, 53, 57, 60, 61, 62, 64, 68,
       72])
../_images/notebooks_scar_robustness_43_1.png 
[106]:

evecs_df.iloc[44]

[106]:

eval    4.678363
kin     0.730499
pot     3.947864
Name: 44, dtype: float64

[130]:

coup_j = np.ones((np.prod(model.shape), 1))
coup_j[0] = 1.05
energy_lump_model = QuantumLinkModel(coup_j, coup_rk, model.shape, basis)
energy_lump_ham = energy_lump_model.hamiltonian

# scar_state = np.zeros((basis.n_states, 1))
# scar_state[np.array([30, 45, 49, 26, 29, 44, 15, 52, 60, 17, 23, 62]), 0] = np.array([-1, -1, -1, 1, 1, 1, 1, 1, 1, -1, -1, -1])  # based on 44
# scar_state[np.array([3, 11, 16, 32, 36, 47, 58, 69]), 0] = np.array([1, -1, -1, 1, 1, -1, 1, -1])  # based on 24
# scar_state /= np.linalg.norm(scar_state)
scar_state = evecs[:, 44]

start, stop, num = 0, 2000, 100
evol_states = expm_multiply(-1j * energy_lump_ham, scar_state, start=start, stop=stop, num=num)
fidelity = [np.abs(evol_state.T @ scar_state).item() ** 2 for evol_state in evol_states]
entropy = [model.entropy(evol_state.flatten(), 1, 0) for evol_state in evol_states]

plt.plot(np.linspace(start, stop, num), fidelity, linestyle="--", marker="o")
plt.ylim(0.0001, 1.1)
# plt.yscale('log')
plt.xlabel("t")
plt.ylabel("Fidelity(t)")
plt.tight_layout()
plt.show()

plt.plot(np.linspace(start, stop, num), entropy, linestyle="--", marker="o")
plt.xlabel("t")
plt.ylabel("EE(t)")
plt.tight_layout()
plt.show()

final_state = np.real_if_close(evol_states[-1], tol=1e-12)
print(f"final fidelity = {fidelity[-1]}")
print(f"O_kin @ psi_scar = {(model.kinetic_term @ final_state).flatten()}")
print(f"<O_kin> = {(final_state.T @ model.kinetic_term @ final_state).item()}")
print(f"<O_pot> = {(final_state.T @ model.potential_term @ final_state).item()}")
print(f"<H> = {(final_state.T @ model.hamiltonian @ final_state).item()}")
../_images/notebooks_scar_robustness_45_0.png 
final fidelity = 0.9684200052672003
O_kin @ psi_scar = [ 4.74315894e-03-0.01351011j  1.03124779e-02+0.0318116j
 -7.33306781e-03-0.03792991j  1.52997953e-02+0.18422271j
  1.03124779e-02+0.0318116j   5.65523991e-03-0.00780859j
 -2.38554269e-03+0.01385663j -3.13183015e-03+0.0050453j
 -7.33306781e-03-0.03792991j -2.38554269e-03+0.01385663j
 -8.56180912e-03-0.00084907j -1.06443298e-02-0.21186007j
  3.77536184e-03+0.01009899j -6.65967834e-02-0.23316644j
  4.68541366e-03+0.00990252j  4.80413953e-02+0.15468343j
  8.00044595e-02+0.21580224j -5.40763198e-02-0.1522819j
 -1.46383257e-02-0.00309953j  2.79780403e-03-0.00045922j
  8.00265490e-03+0.03862601j -4.03437097e-03+0.01099935j
 -8.74394704e-03+0.02450193j -6.58516552e-03-0.12468163j
 -4.03437097e-03+0.01099935j -5.63272425e-04+0.00214595j
  4.15657095e-02+0.13874161j -8.01670083e-03+0.00049375j
 -8.74394704e-03+0.02450193j  4.15657095e-02+0.13874161j
 -2.04103940e-02-0.15438129j -1.65784189e-02-0.00708621j
  4.32299982e-02+0.22165957j -1.01782856e-02+0.00430125j
 -5.06235089e-03+0.00535867j  3.50300113e-03-0.0122706j
 -5.43469476e-02-0.20895467j -4.47406122e-03-0.03199298j
 -1.39877506e-02-0.03288221j -4.09086539e-03-0.01194989j
 -1.99571287e-03-0.01061221j -7.85315053e-03+0.0032013j
  1.65833918e-02+0.01981576j -4.09086539e-03-0.01194989j
  3.82538196e-02+0.14726209j -3.61579908e-02-0.15076817j
  5.70365967e-02+0.21604356j -4.83700054e-02-0.2317743j
 -1.99571287e-03-0.01061221j -3.61579908e-02-0.15076817j
 -4.72334768e-03+0.00378277j  5.41275120e-05+0.00747382j
  4.61661124e-03+0.1427896j   1.67261583e-02-0.01445576j
  3.83624394e-03-0.00639957j  3.94113378e-03+0.03493185j
  8.00044595e-02+0.21580224j -5.06235089e-03+0.00535867j
 -6.65967834e-02-0.23316644j  4.68541366e-03+0.00990252j
  4.80413953e-02+0.15468343j -1.01782856e-02+0.00430125j
 -5.40763198e-02-0.1522819j   3.50300113e-03-0.0122706j
  2.79780403e-03-0.00045922j -4.47406122e-03-0.03199298j
 -1.46383257e-02-0.00309953j -5.43469476e-02-0.20895467j
 -7.85315053e-03+0.0032013j   5.70365967e-02+0.21604356j
  5.41275120e-05+0.00747382j  1.99522153e-02+0.04243186j
  1.67261583e-02-0.01445576j -1.21787411e-02-0.03798336j
  3.94113378e-03+0.03493185j -1.92233570e-03+0.01589628j]
<O_kin> = (-0.6240604520813793+0.3289334174913119j)
<O_pot> = (-3.4166034229539792+1.73700287895565j)
<H> = (-4.040663875035359+2.065936296446962j)

QDM Lattice 6x4#

[28]:

coup_j, coup_rk = (1, 1)
basis, model = setup_dimer_model(lattice_shape=(6, 4), n_solution=1456, coup_j=coup_j, coup_rk=coup_rk)
evals, evecs = np.linalg.eigh(model.hamiltonian)

2023-12-23 21:51:23 [deep_first_search.py] INFO: Deep First Search starts.
2023-12-23 21:51:32 [deep_first_search.py] INFO: Found 1456 Solutions as required in 27921 steps.

[29]:

df = basis.dataframe.copy(deep=True)
df.index = format_custom_index(df.index)
df

[29]:
0 1 2 3 4 5 6 7 8 9 ... 38 39 40 41 42 43 44 45 46 47
(0) 75046264433595 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(1) 75046599846843 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(2) 75051631045563 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(3) 75051966458811 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(4) 75053308111803 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(5) 75053308439355 0 1 0 0 0 1 0 0 0 1 ... 1 1 0 0 1 1 1 0 1 1
(6) 75063461071803 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(7) 75063796485051 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(8) 75063880338363 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(9) 75063880358835 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 0 0 1 1
(10) 75068827683771 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(11) 75069163097019 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(12) 75069246950331 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(13) 75069246970803 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 0 0 1 1
(14) 75070504750011 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(15) 75070505077563 0 1 0 0 0 1 0 0 0 1 ... 1 1 0 0 1 1 1 0 1 1
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(17) 75070588623795 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 0 0 1 1
(18) 75070588930875 0 1 0 0 0 1 0 0 0 1 ... 1 1 0 0 1 1 1 0 1 1
(19) 75070588951347 0 1 0 0 0 1 0 0 0 1 ... 1 1 0 0 1 1 0 0 1 1
(20) 75070589033235 0 1 0 0 0 1 0 0 0 1 ... 1 1 0 0 0 1 0 0 1 1
(21) 75090294131643 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(22) 75090299372475 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(23) 75090629544891 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(24) 75090634785723 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(25) 75090713398203 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
(26) 75090713418675 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 0 0 1 1
(27) 75090718639035 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
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(30) 75091971197883 0 1 0 0 0 1 0 0 0 1 ... 1 1 1 0 1 1 1 0 1 1
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1456 rows × 48 columns

[30]:

evecs_df = pd.DataFrame.from_dict(
    {
        "eval": evals,
        "kin": [(evec.T @ model.kinetic_term @ evec).item() for evec in evecs.T],
        "pot": [(evec.T @ model.potential_term @ evec).item() for evec in evecs.T],
    }
)
evecs_df[evecs_df["kin"].abs() < 1e-12]

[30]:
eval kin pot 
[ ]:

k_rank = np.linalg.matrix_rank(model.kinetic_term)
h_rank = np.linalg.matrix_rank(model.hamiltonian)
print(k_rank, h_rank, basis.n_states, (basis.n_states - k_rank) / basis.n_states)

[32]:

g = nx.from_numpy_array(-model.kinetic_term)
# left, right = nx.bipartite.sets(g)
# pos = nx.bipartite_layout(g, left, align='horizontal')
# pos = nx.spectral_layout(g)
# pos = {k: v + np.random.normal(0, 0.08, 2) for k, v in pos.items()}
# fig = plt.figure(figsize=(82, 32))
# nx.draw(g, pos, with_labels=True, node_size=5000, font_size=42, arrows=True, connectionstyle="arc3,rad=0.15")
# plt.show()

plt.spy(evecs, precision=1e-12, markersize=0.01)
plt.show()
../_images/notebooks_scar_robustness_51_0.png 
[53]:

net = nt.Network(notebook=True, filter_menu=True, select_menu=True, font_color="black")
net.from_nx(g)
for node in net.nodes:
    node["title"] = str(node["id"])
net.show_buttons(filter_=['nodes', 'edges', 'physics'])
net.show("nx.html")

Warning: When  cdn_resources is 'local' jupyter notebook has issues displaying graphics on chrome/safari. Use cdn_resources='in_line' or cdn_resources='remote' if you have issues viewing graphics in a notebook.
nx.html

[53]:

[58]:

ig = igraph.Graph.from_networkx(g)
ig.vs["label"] = [str(i) for i in range(ig.vcount())]
color = ["orange" for i in range(ig.vcount())]
highlight = [
    [1099, 1102, 1239, 1256],
    [604, 601, 331, 311],
    [830, 891, 205, 117],
    [355, 868, 808, 352],
    [390, 367, 655, 695],
    [1047, 1212, 1226, 1082]
]
highlight_color = ["aqua", "deepskyblue", "yellowgreen", "royalblue", "blueviolet", "cornflowerblue"]
for i, nodes in enumerate(highlight):
    for node in nodes:
        color[node] = highlight_color[i]
ig.vs["color"] = color
# fig, ax = plt.subplots(figsize=(10, 10), facecolor="white")
igraph.plot(
    ig,
    layout=ig.layout_kamada_kawai(),
    vertex_size=8,
    # vertex_color="orange",
    edge_width=0.4,
    # edge_color="darkgray",
    vertex_label_size=6,
    # target=ax
)

[58]:
../_images/notebooks_scar_robustness_53_0.svg
[136]:

entropies = np.asarray([model.entropy(evecs[:, i], 0, 0) for i in range(basis.n_states)])
entropy_df = pd.Series(entropies, name="entropy")
entropy_df[entropy_df < 1.6]

[136]:

254     0.768489
1219    1.511229
Name: entropy, dtype: float64

[137]:

entropy_df.plot(style='o')

[137]:

<Axes: >
../_images/notebooks_scar_robustness_55_1.png 
[139]:

plt.plot(evecs[:, 254], linestyle='--', marker='o')
np.where(np.abs(evecs[:, 254]) > 0.15)[0]

[139]:

array([ 367,  390,  655,  695, 1099, 1102, 1239, 1256])
../_images/notebooks_scar_robustness_56_1.png 
[62]:

fig, axes = plt.subplots(2, 2, figsize=(16, 12), facecolor="white")
ax = axes.flatten()

for i, val in enumerate([891, 829, 1213, 354]):  #[1099, 1102, 1239, 1256]
    g = GraphVisualizer(SquareLattice(*model.shape, basis.dataframe.iloc[val]))
    g.plot(show=False, ax=ax[i], node_size=800)
    ax[i].set_title(f"{df.index[val]}")
../_images/notebooks_scar_robustness_57_0.png 
[113]:

g = GraphVisualizer(SquareLattice(*model.shape, basis.dataframe.iloc[1213]))
g.plot(node_size=800)
../_images/notebooks_scar_robustness_58_0.png 
[42]:

coup_j = np.ones((np.prod(model.shape), 1))
coup_j[0] = 1.1
energy_lump_model = QuantumLinkModel(coup_j, coup_rk, model.shape, basis)
energy_lump_ham = energy_lump_model.hamiltonian

scar_state = np.zeros((basis.n_states, 1))
scar_state[np.array([1099, 1102, 1239, 1256]), 0] = np.array([1, -1, -1, 1])
scar_state /= np.linalg.norm(scar_state)

start, stop, num = 0, 200, 100
evol_states = expm_multiply(-1j * energy_lump_ham, scar_state, start=start, stop=stop, num=num)
fidelity = [np.abs(evol_state.T @ scar_state).item() ** 2 for evol_state in evol_states]
entropy = [model.entropy(evol_state.flatten(), 1, 0) for evol_state in evol_states]

plt.plot(np.linspace(start, stop, num), fidelity, linestyle="--", marker="o")
plt.ylim(0.0001, 1.1)
# plt.yscale('log')
plt.xlabel("t")
plt.ylabel("Fidelity(t)")
plt.tight_layout()
plt.show()

plt.plot(np.linspace(start, stop, num), entropy, linestyle="--", marker="o")
plt.xlabel("t")
plt.ylabel("EE(t)")
plt.tight_layout()
plt.show()

final_state = np.real_if_close(evol_states[-1], tol=1e-12)
print(f"final fidelity = {fidelity[-1]}")
print(f"O_kin @ psi_scar = {(model.kinetic_term @ final_state).flatten()}")
print(f"<O_kin> = {(final_state.T @ model.kinetic_term @ final_state).item()}")
print(f"<O_pot> = {(final_state.T @ model.potential_term @ final_state).item()}")
print(f"<H> = {(final_state.T @ model.hamiltonian @ final_state).item()}")

/Users/tandaolin/projects/qlinks/.venv/lib/python3.11/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.py:1272: RuntimeWarning: k >= N - 1 for N * N square matrix. Attempting to use scipy.linalg.eig instead.
  warnings.warn("k >= N - 1 for N * N square matrix. "
../_images/notebooks_scar_robustness_59_1.png
../_images/notebooks_scar_robustness_59_2.png 
final fidelity = 1.000000000000004
O_kin @ psi_scar = [0.+0.j 0.+0.j 0.+0.j ... 0.+0.j 0.+0.j 0.+0.j]
<O_kin> = 0j
<O_pot> = (-2.3934538551791857+3.204899162708247j)
<H> = (-2.3934538551791857+3.204899162708247j)

QDM Lattice 8x4#

[94]:

coup_j, coup_rk = (1, 1)  # dfs 2 mins 50 secs
basis, model = setup_dimer_model(lattice_shape=(8, 4), n_solution=17412, coup_j=coup_j, coup_rk=coup_rk)

2023-12-23 22:58:43 [deep_first_search.py] INFO: Deep First Search starts.
2023-12-23 23:01:16 [deep_first_search.py] INFO: Found 17412 Solutions as required in 479085 steps.

[104]:

# basis.dataframe.to_parquet("qdm_8x4_lattice.parquet", index=False)
basis = ComputationBasis.from_parquet("qdm_8x4_lattice.parquet")
model = QuantumLinkModel(coup_j, coup_rk, (8, 4), basis)

[106]:

evals, evecs = eigsh(csr_matrix(model.hamiltonian), k=1000, which="SA")

---------------------------------------------------------------------------
KeyboardInterrupt                         Traceback (most recent call last)
Cell In[106], line 1
----> 1 evals, evecs = eigsh(csr_matrix(model.hamiltonian), k=1000, which="SA")

File ~/projects/qlinks/.venv/lib/python3.11/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.py:1697, in eigsh(A, k, M, sigma, which, v0, ncv, maxiter, tol, return_eigenvectors, Minv, OPinv, mode)
   1695 with _ARPACK_LOCK:
   1696     while not params.converged:
-> 1697         params.iterate()
   1699     return params.extract(return_eigenvectors)

File ~/projects/qlinks/.venv/lib/python3.11/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.py:537, in _SymmetricArpackParams.iterate(self)
    535 def iterate(self):
    536     self.ido, self.tol, self.resid, self.v, self.iparam, self.ipntr, self.info = \
--> 537         self._arpack_solver(self.ido, self.bmat, self.which, self.k,
    538                             self.tol, self.resid, self.v, self.iparam,
    539                             self.ipntr, self.workd, self.workl, self.info)
    541     xslice = slice(self.ipntr[0] - 1, self.ipntr[0] - 1 + self.n)
    542     yslice = slice(self.ipntr[1] - 1, self.ipntr[1] - 1 + self.n)

KeyboardInterrupt:

[97]:

df = basis.dataframe.copy(deep=True)
df.index = format_custom_index(df.index)
df

[97]:
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17412 rows × 64 columns

[56]:

g = nx.from_numpy_array(-model.kinetic_term)
ig = igraph.Graph.from_networkx(g)
igraph.plot(
    ig,
    # layout=ig.layout_kamada_kawai(),
    vertex_size=4,
    vertex_color="orange",
    edge_width=0.1,
    edge_color="gray"
)

[56]:
../_images/notebooks_scar_robustness_65_0.svg
[43]:

plt.spy(evecs, precision=1e-12, markersize=0.05)
plt.show()

---------------------------------------------------------------------------
KeyboardInterrupt                         Traceback (most recent call last)
Cell In[43], line 10
      2 # left, right = nx.bipartite.sets(g)
      3 # pos = nx.bipartite_layout(g, left, align='horizontal')
      4 # pos = nx.spectral_layout(g)
   (...)
      7 # nx.draw(g, pos, with_labels=True, node_size=5000, font_size=42, arrows=True, connectionstyle="arc3,rad=0.15")
      8 # plt.show()
      9 ig = igraph.Graph.from_networkx(g)
---> 10 igraph.plot(ig, layout=ig.layout_kamada_kawai(), vertex_size=10, vertex_color="orange")
     11 plt.show()
     13 plt.spy(evecs, precision=1e-12, markersize=0.05)

File ~/projects/qlinks/.venv/lib/python3.11/site-packages/igraph/layout.py:691, in _layout_method_wrapper.<locals>.result(*args, **kwds)
    690 def result(*args, **kwds):
--> 691     layout = func(*args, **kwds)
    692     if not isinstance(layout, Layout):
    693         layout = Layout(layout)

KeyboardInterrupt:

Check effective matrix#

[71]:

l = 0.0
dj = 1.1

mat = np.array(
    [
        [l, dj, dj, 0, 0, 0, 0, 0, 0],
        [dj, l, 0, 1, 1, 0, 0, 0, 0],
        [dj, 0, l, 0, 1, 1, 0, 0, 0],
        [0, 1, 0, l, 0, 0, 1, 0, 0],
        [0, 1, 1, 0, l, 0, 1, 1, 0],
        [0, 0, 1, 0, 0, l, 0, 1, 0],
        [0, 0, 0, 1, 1, 0, l, 0, 1],
        [0, 0, 0, 0, 1, 1, 0, l, 1],
        [0, 0, 0, 0, 0, 0, 1, 1, l]
    ]
)

np.linalg.matrix_power(mat, 2) @ np.array([[0, 1, -1, 0, 0, 0, -1, 1, 0]]).T

[71]:

array([[0.],
       [0.],
       [0.],
       [0.],
       [0.],
       [0.],
       [0.],
       [0.],
       [0.]])

[75]:

from IPython.display import clear_output
from time import sleep

for i in range(20):
    print(f"matrix power {i+1}")
    walker = np.linalg.matrix_power(mat, i+1)
    walker /= np.linalg.norm(walker)
    plt.matshow(walker, animated=True)
    plt.colorbar()
    plt.show()
    sleep(0.3)
    clear_output(wait=True)

matrix power 20
../_images/notebooks_scar_robustness_69_1.png

QLM Lattice 4x2#

[5]:

coup_j, coup_rk = (1, 1)
basis, model = setup_link_model(lattice_shape=(4, 2), n_solution=38, coup_j=coup_j, coup_rk=coup_rk)
evals, evecs = np.linalg.eigh(model.hamiltonian)

2023-12-23 23:44:19 [deep_first_search.py] INFO: Deep First Search starts.
2023-12-23 23:44:19 [deep_first_search.py] INFO: Found 38 Solutions as required in 231 steps.

[6]:

df = basis.dataframe.copy(deep=True)
df.index = format_custom_index(df.index)
df

[6]:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
(0) 1455 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1
(1) 4539 0 0 0 1 0 0 0 1 1 0 1 1 1 0 1 1
(2) 5310 0 0 0 1 0 1 0 0 1 0 1 1 1 1 1 0
(3) 6582 0 0 0 1 1 0 0 1 1 0 1 1 0 1 1 0
(4) 9627 0 0 1 0 0 1 0 1 1 0 0 1 1 0 1 1
(5) 11670 0 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0
(6) 16875 0 1 0 0 0 0 0 1 1 1 1 0 1 0 1 1
(7) 17646 0 1 0 0 0 1 0 0 1 1 1 0 1 1 1 0
(8) 18093 0 1 0 0 0 1 1 0 1 0 1 0 1 1 0 1
(9) 18918 0 1 0 0 1 0 0 1 1 1 1 0 0 1 1 0
(10) 19365 0 1 0 0 1 0 1 1 1 0 1 0 0 1 0 1
(11) 20730 0 1 0 1 0 0 0 0 1 1 1 1 1 0 1 0
(12) 21177 0 1 0 1 0 0 1 0 1 0 1 1 1 0 0 1
(13) 23220 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0 0
(14) 25818 0 1 1 0 0 1 0 0 1 1 0 1 1 0 1 0
(15) 26265 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1
(16) 27090 0 1 1 0 1 0 0 1 1 1 0 1 0 0 1 0
(17) 27537 0 1 1 0 1 0 1 1 1 0 0 1 0 0 0 1
(18) 28308 0 1 1 0 1 1 1 0 1 0 0 1 0 1 0 0
(19) 37227 1 0 0 1 0 0 0 1 0 1 1 0 1 0 1 1
(20) 37998 1 0 0 1 0 1 0 0 0 1 1 0 1 1 1 0
(21) 38445 1 0 0 1 0 1 1 0 0 0 1 0 1 1 0 1
(22) 39270 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0
(23) 39717 1 0 0 1 1 0 1 1 0 0 1 0 0 1 0 1
(24) 42315 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1
(25) 44358 1 0 1 0 1 1 0 1 0 1 0 0 0 1 1 0
(26) 44805 1 0 1 0 1 1 1 1 0 0 0 0 0 1 0 1
(27) 46170 1 0 1 1 0 1 0 0 0 1 0 1 1 0 1 0
(28) 46617 1 0 1 1 0 1 1 0 0 0 0 1 1 0 0 1
(29) 47442 1 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0
(30) 47889 1 0 1 1 1 0 1 1 0 0 0 1 0 0 0 1
(31) 48660 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 0
(32) 53865 1 1 0 1 0 0 1 0 0 1 1 0 1 0 0 1
(33) 55908 1 1 0 1 1 0 1 0 0 1 1 0 0 1 0 0
(34) 58953 1 1 1 0 0 1 1 0 0 1 0 0 1 0 0 1
(35) 60225 1 1 1 0 1 0 1 1 0 1 0 0 0 0 0 1
(36) 60996 1 1 1 0 1 1 1 0 0 1 0 0 0 1 0 0
(37) 64080 1 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0 
[35]:

evecs_df = pd.DataFrame.from_dict(
    {
        "eval": evals,
        "kin": [(evec.T @ model.kinetic_term @ evec).item() for evec in evecs.T],
        "pot": [(evec.T @ model.potential_term @ evec).item() for evec in evecs.T],
    }
)
evecs_df[evecs_df["kin"].abs() < 1e-12]

[35]:
eval kin pot
16 4.0 -6.434147e-30 4.0
18 4.0 -4.659210e-30 4.0
19 4.0 -5.482583e-29 4.0
20 4.0 -1.774937e-30 4.0
21 5.0 4.240127e-30 5.0
22 5.0 3.056836e-29 5.0
23 5.0 2.465190e-30 5.0
24 5.0 6.064368e-30 5.0 
[36]:

k_rank = np.linalg.matrix_rank(model.kinetic_term)
h_rank = np.linalg.matrix_rank(model.hamiltonian)
print(k_rank, h_rank, basis.n_states, (basis.n_states - k_rank) / basis.n_states)

24 37 38 0.3684210526315789

[43]:

plt.spy(evecs, precision=1e-12, markersize=5.0)
plt.show()
../_images/notebooks_scar_robustness_75_0.png 
[46]:

g = nx.from_numpy_array(-model.kinetic_term)
ig = igraph.Graph.from_networkx(g)
ig.vs["label"] = [str(i) for i in range(ig.vcount())]
igraph.plot(
    ig,
    layout=ig.layout_kamada_kawai(),
    vertex_size=16,
    vertex_color="orange",
)

[46]:
../_images/notebooks_scar_robustness_76_0.svg
[52]:

entropies = np.asarray([model.entropy(evecs[:, i], 1, 0) for i in range(basis.n_states)])
entropy_df = pd.Series(entropies, name="entropy")
entropy_df[entropy_df < 1.8]

[52]:

7     1.656653
8     1.641885
18    1.795919
19    1.696326
29    1.655564
36    1.342927
37    1.576712
Name: entropy, dtype: float64

[51]:

entropy_df.plot(style='o')

[51]:

<Axes: >
../_images/notebooks_scar_robustness_78_1.png 
[53]:

i = 16
plt.plot(evecs[:, i], linestyle='--', marker='o')
np.where(np.abs(evecs[:, i]) > 0.15)[0]

[53]:

array([ 1, 10, 13, 16, 21, 24, 27, 36])
../_images/notebooks_scar_robustness_79_1.png 
[57]:

fig, axes = plt.subplots(10, 4, figsize=(18, 40), facecolor="white")
ax = axes.flatten()

for i in range(basis.n_states):
    g = GraphVisualizer(SquareLattice(*model.shape, basis.dataframe.iloc[i]))
    g.plot(show=False, ax=ax[i], node_size=800)
    ax[i].set_title(f"{df.index[i]}")
../_images/notebooks_scar_robustness_80_0.png 
[8]:

coup_j = np.ones((np.prod(model.shape), 1))
coup_j[1] = 1.5
energy_lump_model = QuantumLinkModel(coup_j, coup_rk, model.shape, basis)
energy_lump_ham = energy_lump_model.hamiltonian

scar_state = np.zeros((basis.n_states, 1))
scar_state[np.array([1, 13, 24, 36]), 0] = np.array([-1, 1, 1, -1])
scar_state /= np.linalg.norm(scar_state)

start, stop, num = 0, 1000, 100
evol_states = expm_multiply(-1j * energy_lump_ham, scar_state, start=start, stop=stop, num=num)
fidelity = [np.abs(evol_state.T @ scar_state).item() ** 2 for evol_state in evol_states]
entropy = [model.entropy(evol_state.flatten(), 1, 0) for evol_state in evol_states]

plt.plot(np.linspace(start, stop, num), fidelity, linestyle="--", marker="o")
plt.ylim(0.0001, 1.1)
# plt.yscale('log')
plt.xlabel("t")
plt.ylabel("Fidelity(t)")
plt.tight_layout()
plt.show()

plt.plot(np.linspace(start, stop, num), np.round(entropy, 12), linestyle="--", marker="o")
plt.xlabel("t")
plt.ylabel("EE(t)")
plt.tight_layout()
plt.show()

final_state = np.real_if_close(evol_states[-1], tol=1e-12)
print(f"final fidelity = {fidelity[-1]}")
print(f"O_kin @ psi_scar = {(model.kinetic_term @ final_state).flatten()}")
print(f"<O_kin> = {(final_state.T @ model.kinetic_term @ final_state).item()}")
print(f"<O_pot> = {(final_state.T @ model.potential_term @ final_state).item()}")
print(f"<H> = {(final_state.T @ model.hamiltonian @ final_state).item()}")

/Users/tandaolin/projects/qlinks/.venv/lib/python3.11/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.py:1272: RuntimeWarning: k >= N - 1 for N * N square matrix. Attempting to use scipy.linalg.eig instead.
  warnings.warn("k >= N - 1 for N * N square matrix. "
../_images/notebooks_scar_robustness_81_1.png
../_images/notebooks_scar_robustness_81_2.png 
final fidelity = 0.7741120945201118
O_kin @ psi_scar = [-1.26816175e-01+2.16458990e-01j -1.74860126e-15-3.71924713e-15j
 -2.49510615e-15+3.02725097e-17j  4.06474110e-16+1.59762660e-15j
  6.27588676e-02-6.25774945e-02j -1.26737070e-15-7.74890865e-16j
  1.26816175e-01-2.16458990e-01j -1.28369537e-15+8.18789481e-16j
  6.27588676e-02-6.25774945e-02j -6.27588676e-02+6.25774945e-02j
 -5.72458747e-16-9.02056208e-16j -1.88632146e-15-8.22702315e-16j
  8.30786076e-16+4.32674238e-17j -2.58126853e-15-5.27355937e-16j
 -1.53222678e-15+4.74340519e-17j -1.83878645e-15-1.16591307e-15j
 -2.35823835e-15+2.43020098e-16j  8.02828449e-16-6.46296253e-16j
 -6.27588676e-02+6.25774945e-02j -6.27588676e-02+6.25774945e-02j
 -1.28905152e-15+2.05400241e-15j -2.00000525e-15+1.21800850e-15j
 -4.23278892e-15+8.57837956e-16j  1.06105148e-16+8.60137442e-16j
  8.32667268e-16-1.22124533e-15j -2.63677968e-15+5.55111512e-17j
 -9.56214122e-16+3.54749084e-16j -3.78516662e-15+2.35922393e-15j
 -6.27588676e-02+6.25774945e-02j  6.27588676e-02-6.25774945e-02j
 -3.07393000e-15+6.52256027e-16j  1.26816175e-01-2.16458990e-01j
 -4.89783893e-16-9.79329937e-16j  6.27588676e-02-6.25774945e-02j
 -1.24900090e-15-9.43689571e-16j -1.42226886e-16-3.88756955e-16j
  0.00000000e+00+1.97064587e-15j -1.26816175e-01+2.16458990e-01j]
<O_kin> = (-0.04469297354403623-0.17216447669592444j)
<O_pot> = (-2.3013209894209448-2.8002538832799444j)
<H> = (-2.3460139629649808-2.972418359975869j)

QLM Lattice 4x4#

[59]:

coup_j, coup_rk = (1, 1)
basis, model = setup_link_model(lattice_shape=(4, 4), n_solution=990, coup_j=coup_j, coup_rk=coup_rk)
evals, evecs = np.linalg.eigh(model.hamiltonian)

2023-12-23 22:09:14 [deep_first_search.py] INFO: Deep First Search starts.
2023-12-23 22:09:19 [deep_first_search.py] INFO: Found 990 Solutions as required in 9381 steps.

[60]:

df = basis.dataframe.copy(deep=True)
df.index = format_custom_index(df.index)
df

[60]:
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990 rows × 32 columns

[61]:

evecs_df = pd.DataFrame.from_dict(
    {
        "eval": evals,
        "kin": [(evec.T @ model.kinetic_term @ evec).item() for evec in evecs.T],
        "pot": [(evec.T @ model.potential_term @ evec).item() for evec in evecs.T],
    }
)
evecs_df[evecs_df["kin"].abs() < 1e-12]

[61]:
eval kin pot
574 8.0 -1.349265e-18 8.0
575 8.0 1.875301e-18 8.0
576 8.0 4.546683e-18 8.0
577 8.0 5.921708e-19 8.0
578 8.0 5.162581e-18 8.0
579 8.0 -4.928153e-18 8.0
580 8.0 -1.140181e-17 8.0
581 8.0 -2.148948e-18 8.0
582 8.0 -1.139690e-17 8.0
583 8.0 4.990670e-18 8.0
584 8.0 8.727366e-20 8.0
585 8.0 -1.531264e-18 8.0
586 8.0 1.152132e-17 8.0
587 8.0 -2.056418e-18 8.0
588 8.0 1.160960e-17 8.0
589 8.0 8.304220e-18 8.0
590 8.0 -2.619799e-19 8.0
591 8.0 4.393195e-19 8.0
592 8.0 -2.615988e-18 8.0
593 8.0 -3.253685e-18 8.0
594 8.0 2.828691e-18 8.0
595 8.0 -2.132213e-17 8.0
596 8.0 -1.095384e-17 8.0
597 8.0 -5.541874e-18 8.0
598 8.0 -3.299282e-18 8.0
599 8.0 5.322860e-18 8.0 
[62]:

k_rank = np.linalg.matrix_rank(model.kinetic_term)
h_rank = np.linalg.matrix_rank(model.hamiltonian)
print(k_rank, h_rank, basis.n_states, (basis.n_states - k_rank) / basis.n_states)

832 989 990 0.1595959595959596

[67]:

plt.spy(evecs, precision=1e-12, markersize=0.05)
plt.show()
../_images/notebooks_scar_robustness_87_0.png 
[74]:

g = nx.from_numpy_array(-model.kinetic_term)
ig = igraph.Graph.from_networkx(g)
ig.vs["label"] = [str(i) for i in range(ig.vcount())]
igraph.plot(
    ig,
    layout=ig.layout_kamada_kawai(),
    vertex_size=10,
    vertex_color="orange",
    edge_width=0.8,
    # edge_color="gray"
    vertex_label_size=6,
)

[74]:
../_images/notebooks_scar_robustness_88_0.svg
[77]:

entropies = np.asarray([model.entropy(evecs[:, i], 1, 0) for i in range(basis.n_states)])
entropy_df = pd.Series(entropies, name="entropy")
entropy_df[entropy_df < 3.5]

[77]:

12     3.437326
13     3.354152
16     3.485520
17     3.295924
18     3.337060
20     3.121310
29     3.189914
30     3.121809
31     3.227343
32     3.121694
55     3.341194
577    2.857901
579    3.233935
584    3.482694
586    3.285896
767    3.459376
786    3.191232
789    3.288996
842    3.490871
868    3.460691
905    3.426521
951    3.131694
955    3.079945
958    3.378943
959    3.397993
960    3.295655
961    3.222521
962    3.351094
963    3.388660
964    2.999035
965    3.302850
966    3.402129
973    3.164994
974    3.376386
975    3.334039
976    3.404896
977    3.450523
978    3.294743
982    3.410129
988    2.059264
989    2.229480
Name: entropy, dtype: float64

[76]:

entropy_df.plot(style='o')

[76]:

<Axes: >
../_images/notebooks_scar_robustness_90_1.png 
[89]:

i = 574
plt.plot(evecs[:, i], linestyle='--', marker='o')
np.where(np.abs(evecs[:, i]) > 0.09)[0]

[89]:

array([ 35,  82,  90,  91, 140, 183, 216, 226, 257, 299, 307, 331, 333,
       334, 338, 378, 412, 455, 534, 577, 611, 651, 655, 656, 658, 682,
       690, 732, 763, 773, 806, 849, 898, 899, 907, 954])
../_images/notebooks_scar_robustness_91_1.png 
[87]:

fig, axes = plt.subplots(4, 2, figsize=(16, 24), facecolor="white")
ax = axes.flatten()

for i, val in enumerate([35, 82, 299, 333, 656, 690, 732, 773]):
    g = GraphVisualizer(SquareLattice(*model.shape, basis.dataframe.iloc[val]))
    g.plot(show=False, ax=ax[i], node_size=800)
    ax[i].set_title(f"{df.index[val]}")
../_images/notebooks_scar_robustness_92_0.png 
[90]:

coup_j = np.ones((np.prod(model.shape), 1))
# coup_j[1] = 1.1
energy_lump_model = QuantumLinkModel(coup_j, coup_rk, model.shape, basis)
energy_lump_ham = energy_lump_model.hamiltonian

scar_state = np.zeros((basis.n_states, 1))
scar_state[np.array([82, 333]), 0] = np.array([1, -1])
scar_state /= np.linalg.norm(scar_state)

start, stop, num = 0, 200, 100
evol_states = expm_multiply(-1j * energy_lump_ham, scar_state, start=start, stop=stop, num=num)
fidelity = [np.abs(evol_state.T @ scar_state).item() ** 2 for evol_state in evol_states]
entropy = [model.entropy(evol_state.flatten(), 2, 0) for evol_state in evol_states]

plt.plot(np.linspace(start, stop, num), fidelity, linestyle="--", marker="o")
plt.ylim(0.0001, 1.1)
# plt.yscale('log')
plt.xlabel("t")
plt.ylabel("Fidelity(t)")
plt.tight_layout()
plt.show()

plt.plot(np.linspace(start, stop, num), np.round(entropy, 12), linestyle="--", marker="o")
plt.xlabel("t")
plt.ylabel("EE(t)")
plt.tight_layout()
plt.show()

final_state = np.real_if_close(evol_states[0], tol=1e-12)
print(f"final fidelity = {fidelity[0]}")
print(f"O_kin @ psi_scar = {(model.kinetic_term @ final_state).flatten()}")
print(f"<O_kin> = {(final_state.T @ model.kinetic_term @ final_state).item()}")
print(f"<O_pot> = {(final_state.T @ model.potential_term @ final_state).item()}")
print(f"<H> = {(final_state.T @ model.hamiltonian @ final_state).item()}")

/Users/tandaolin/projects/qlinks/.venv/lib/python3.11/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.py:1272: RuntimeWarning: k >= N - 1 for N * N square matrix. Attempting to use scipy.linalg.eig instead.
  warnings.warn("k >= N - 1 for N * N square matrix. "
../_images/notebooks_scar_robustness_93_1.png
../_images/notebooks_scar_robustness_93_2.png 
final fidelity = 0.9999999999999996
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<O_kin> = 0.0
<O_pot> = 7.999999999999998
<H> = 7.999999999999998

[ ]: